Computability of the reduced group C-algebra norm for SL_3(Z)

Determine whether the norm on the reduced group C-algebra C^*_r(\operatorname{SL}_3(\mathbb{Z})) is computable.

Background

The paper proves that the norm on C^*_r(G) is computable for every finitely generated group G with solvable word problem and property RD. Discrete cocompact lattices in \operatorname{SL}_3(\mathbb{R}) satisfy property RD, whereas \operatorname{SL}_3(\mathbb{Z}) does not, so the main theorem does not settle the computability question for this specific group.

The authors explicitly identify the computability of the reduced group C-algebra norm for \operatorname{SL}_3(\mathbb{Z}) as an unresolved question. They also note that the analogous question for the universal group C-algebra C^*(\operatorname{SL}_3(\mathbb{Z})) had previously been posed and remains open.